From the random geometry of conformally invariant systems to the K\"ahler geometry of universal Teichm\"uller space

Abstract

The goal of this expository article is to explain how a fundamental functional on the space of Jordan curves arising from SLE - Loewner energy - is connected to a seemingly far apart subject: the K\"ahler geometry of universal Teichm\"uller space. We provide a background on Loewner energy, the universal Liouville action, and the intuition behind the proof of the identity between them.